Subgroup ($H$) information
| Description: | $C_{42}$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
43 & 42 \\
0 & 43
\end{array}\right), \left(\begin{array}{rr}
1 & 60 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 28 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_2^2\times C_6.D_{28}$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^3\times C_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^4:C_2^3:\GL(3,2)$, of order \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2^4:C_2^3:\GL(3,2)$, of order \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_2^4.C_7.C_3^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{W}$ | \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2^3\times C_{42}$ | ||
| Normalizer: | $C_2^2\times C_6.D_{28}$ | ||
| Minimal over-subgroups: | $C_2\times C_{42}$ | $C_2\times C_{42}$ | $C_3\times D_{14}$ |
| Maximal under-subgroups: | $C_{21}$ | $C_{14}$ | $C_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |